Here we only prove the function space case because the sequence space case uses the same argument. The implications (ii) \(\Rightarrow \)  (iii) \(\Rightarrow \)  (iv) , (v) \(\Rightarrow \)  (vi) \(\Rightarrow \)  (vii), and (viii) \(\Rightarrow \)  (ix) \(\Rightarrow \)  (x) are well-known facts [1]}, [2]}. We have (i) \(\Rightarrow \)  (ii) and (i) \(\Rightarrow \)  (v) from Theorem REF . Now, suppose that \(\varphi \)  is an N-function at infinity and satisfies the appropriate \(\Delta _2\) -condition. Then the Orlicz-Lorentz space \(\Lambda _{\varphi ,w} = (\Lambda _{\varphi ,w})_a\)  has the Radon-Nikodým property by Theorem REF . Hence we see that (iv) \(\Rightarrow \)  (i) and (vii) \(\Rightarrow \)  (i). By the duality result between D2Ps and octahedralities, the space \((\Lambda _{\varphi ,w})_a\)  has the SD2P (resp. D2P, LD2P) if and only if \((\Lambda _{\varphi ,w})_a^* \simeq (\Lambda _{\varphi ,w})_a^{\prime } = \mathcal {M}_{\varphi _*,w}^0\)  is octahedral (resp. weakly octahedral, locally octahedral) [2]}.
